# Question about the gaps between prime numbers

• I
Is there any prime number pn, such that it has a relationship with the next prime number pn+1
$$p_{n+1} > p_{n}^2$$
If not, is there any proof saying a prime like this does not exist?

$$p_{n+1} > 2p_{n}$$

fresh_42
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2021 Award
Is there any prime number pn, such that it has a relationship with the next prime number pn+1
$$p_{n+1} > p_{n}^2$$
If not, is there any proof saying a prime like this does not exist?

$$p_{n+1} > 2p_{n}$$
https://en.wikipedia.org/wiki/Prime_gap

There is also a proof for arbitrary gaps, but see the section "upper bounds".

Interesting.Bertrand's Postulate answers the second part of my question. :)

I see Firoozbakht's conjecture, which is similar to my first part, but it's not quite the same thing as
$$p_{n+1} > p_{n}^2$$

I wonder if this can be proved or disproved from other postulates...

micromass
Staff Emeritus
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Interesting.Bertrand's Postulate answers the second part of my question. :)

I see Firoozbakht's conjecture, which is similar to my first part, but it's not quite the same thing as
$$p_{n+1} > p_{n}^2$$

I wonder if this can be proved or disproved from other postulates...

This also follows very easily from Bertrand's postulate.

Stephen Tashi
but it's not quite the same thing as
$$p_{n+1} > p_{n}^2$$

I wonder if this can be proved or disproved from other postulates...

Compare ##2p_n## to ##p^2_n## .

This also follows very easily from Bertrand's postulate.

Yeah it does. Wow. I'm dumb. :p